Editing Banach space (section)

====Examples of dual spaces====

The dual of <math>c_0</math> is isometrically isomorphic to <math>\ell^1</math>: for every bounded linear functional <math>f</math> on <math>c_0,</math> there is a unique element <math>y = \{y_n\} \in \ell^1</math> such that
<math display=block>f(x) = \sum_{n \in \N} x_n y_n, \qquad x = \{x_n\} \in c_0, \ \ \text{and} \ \ \|f\|_{(c_0)'} = \|y\|_{\ell_1}. </math>

The dual of <math>\ell^1</math> is isometrically isomorphic to <math>\ell^{\infty}</math>. 
The dual of [[Lp space#Properties of Lp spaces|Lebesgue space]] <math>L^p([0, 1])</math> is isometrically isomorphic to <math>L^q([0, 1])</math> when <math>1 \leq p < \infty</math> and <math>\frac{1}{p} + \frac{1}{q} = 1.</math>

For every vector <math>y</math> in a Hilbert space <math>H,</math> the mapping
<math display=block>x \in H \to f_y(x) = \langle x, y \rangle</math>

defines a continuous linear functional <math>f_y</math> on <math>H.</math>The [[Riesz representation theorem]] states that every continuous linear functional on <math>H</math> is of the form <math>f_y</math> for a uniquely defined vector <math>y</math> in <math>H.</math>
The mapping <math>y \in H \to f_y</math> is an [[Antilinear map|antilinear]] isometric bijection from <math>H</math> onto its dual <math>H'.</math> 
When the scalars are real, this map is an isometric isomorphism.

When <math>K</math> is a compact Hausdorff topological space, the dual <math>M(K)</math> of <math>C(K)</math> is the space of [[Radon measure]]s in the sense of Bourbaki.<ref>see N. Bourbaki, (2004), "Integration I", Springer Verlag, {{ISBN|3-540-41129-1}}.</ref> 
The subset <math>P(K)</math> of <math>M(K)</math> consisting of non-negative measures of mass 1 ([[probability measure]]s) is a convex w*-closed subset of the unit ball of <math>M(K).</math> 
The [[extreme point]]s of <math>P(K)</math> are the [[Dirac measure]]s on <math>K.</math> 
The set of Dirac measures on <math>K,</math> equipped with the w*-topology, is [[Homeomorphism|homeomorphic]] to <math>K.</math>

{{math theorem|name=[[Banach–Stone theorem|Banach–Stone Theorem]]|math_statement=If <math>K</math> and <math>L</math> are compact Hausdorff spaces and if <math>C(K)</math> and <math>C(L)</math> are isometrically isomorphic, then the topological spaces <math>K</math> and <math>L</math> are [[homeomorphic]].<ref name= Eilenberg /><ref>see also {{harvtxt|Banach|1932}}, p.&nbsp;170 for metrizable <math>K</math> and <math>L.</math></ref>}}

The result has been extended by Amir<ref>{{cite journal |first=Dan |last=Amir |title=On isomorphisms of continuous function spaces |journal=[[Israel Journal of Mathematics]] |volume=3 |year=1965 |issue=4 |pages=205–210 |doi=10.1007/bf03008398 |doi-access=free |s2cid=122294213 }}</ref> and Cambern<ref>{{cite journal |first=M. |last=Cambern |title=A generalized Banach–Stone theorem |journal=Proc. Amer. Math. Soc. |volume=17 |year=1966 |issue=2 |pages=396–400 |doi=10.1090/s0002-9939-1966-0196471-9|doi-access=free}} And {{cite journal |first=M. |last=Cambern |title=On isomorphisms with small bound |journal=Proc. Amer. Math. Soc. |volume=18 |year=1967 |issue=6 |pages=1062–1066 |doi=10.1090/s0002-9939-1967-0217580-2|doi-access=free}}</ref> to the case when the multiplicative [[Banach–Mazur compactum|Banach–Mazur distance]] between <math>C(K)</math> and <math>C(L)</math> is <math>< 2.</math> 
The theorem is no longer true when the distance is <math> = 2.</math><ref>{{cite journal |first=H. B. |last=Cohen |title=A bound-two isomorphism between <math>C(X)</math> Banach spaces |journal=Proc. Amer. Math. Soc. |volume=50 |year=1975 |pages=215–217 |doi=10.1090/s0002-9939-1975-0380379-5|doi-access=free }}</ref>

In the commutative [[Banach algebra]] <math>C(K),</math> the [[Banach algebra#Ideals and characters|maximal ideals]] are precisely kernels of Dirac measures on <math>K,</math>
<math display=block>I_x = \ker \delta_x = \{f \in C(K) \mid f(x) = 0\}, \quad x \in K.</math>

More generally, by the [[Gelfand–Mazur theorem]], the maximal ideals of a unital commutative Banach algebra can be identified with its [[Banach algebra#Ideals and characters|characters]]—not merely as sets but as topological spaces: the former with the [[hull-kernel topology]] and the latter with the w*-topology. 
In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual <math>A'.</math>

{{math theorem|math_statement= If <math>K</math> is a compact Hausdorff space, then the maximal ideal space <math>\Xi</math> of the Banach algebra <math>C(K)</math> is [[homeomorphic]] to <math>K.</math><ref name=Eilenberg>{{cite journal |last=Eilenberg |first=Samuel |title=Banach Space Methods in Topology |journal=[[Annals of Mathematics]] |date=1942 |volume=43 |issue=3 |pages=568–579 |doi=10.2307/1968812|jstor=1968812 }}</ref>}}

Not every unital commutative Banach algebra is of the form <math>C(K)</math> for some compact Hausdorff space <math>K.</math> However, this statement holds if one places <math>C(K)</math> in the smaller category of commutative [[C*-algebra]]s. 
[[Israel Gelfand|Gelfand's]] [[Gelfand representation|representation theorem]] for commutative C*-algebras states that every commutative unital ''C''*-algebra <math>A</math> is isometrically isomorphic to a <math>C(K)</math> space.<ref>See for example {{cite book |first=W. |last=Arveson |year=1976 |title=An Invitation to C*-Algebra |publisher=Springer-Verlag |isbn=0-387-90176-0 }}</ref> 
The Hausdorff compact space <math>K</math> here is again the maximal ideal space, also called the [[Spectrum of a C*-algebra#Examples|spectrum]] of <math>A</math> in the C*-algebra context.